Various sharp estimates for semi-discrete Riesz transforms of the second order
aa r X i v : . [ m a t h . C A ] J a n VARIOUS SHARP ESTIMATES FOR SEMI-DISCRETERIESZ TRANSFORMS OF THE SECOND ORDER
K. DOMELEVO, A. OSE¸KOWSKI, AND S. PETERMICHL
Abstract.
We give several sharp estimates for a class of combinationsof second order Riesz transforms on Lie groups G = G x × G y thatare multiply connected, composed of a discrete abelian component G x and a connected component G y endowed with a biinvariant measure.These estimates include new sharp L p estimates via Choi type constants,depending upon the multipliers of the operator. They also include weak-type, logarithmic and exponential estimates. We give an optimal L q → L p estimate as well.It was shown recently by Arcozzi - Domelevo - Petermichl that suchsecond order Riesz transforms applied to a function may be written asconditional expectation of a simple transformation of a stochastic inte-gral associated with the function.The proofs of our theorems combine this stochastic integral represen-tation with a number of deep estimates for pairs of martingales understrong differential subordination by
Choi , Banuelos and
Ose¸kowski .When two continuous directions are available, sharpness is shown viathe laminates technique. We show that sharpness is preserved in thediscrete case using Lax-Richtmyer theorem. Introduction
Sharp, classical L p norm inequalities for pairs of differentially subordinatemartingales date back to the celebrated work of Burkholder [15] in 1984where the optimal constant is exhibited. See also from the same author[17][18]. The relation between differentially subordinate martingales andCZ (i.e. Cald´eron–Zygmund) operators is known at least since
Gundy – Varopoulos [32].
Banuelos – Wang [12] were the first to exploit thisconnection to prove new sharp inequalities for singular intergrals. This in-tersection of probability theory with classical questions in harmonic analysishas lead to much interest and a vast literature has been accumulating onthis line of research.In this article we state a number of sharp estimates that hold in the veryrecent, new direction concerning the semi–discrete setting, applying it to afamily of second order Riesz transforms on multiply–connected Lie groups.
Date : July 12, 2018.Adam Osekowski is supported by Narodowe Centrum Nauki Poland (NCN), grantDEC-2014/14/E/ST1/00532.Stefanie Petermichl is supported by ERC project CHRiSHarMa DLV-862402.
We recall their representation through stochastic integrals using jump pro-cesses on multiply–connected Lie groups from [3]. In this representationformula jump processes play a role, but the strong differential subordina-tion holds between the martingales representing the test function and theoperator applied to the test function.The usual procedure for obtaining (sharp) inequalities for operators ofCalder´on–Zygmund type from inequalities for martingales is the following.Starting with a test function f , martingales are built using Brownian mo-tion or background noise and harmonic functions in the upper half space R + × R n . Through the use of Itøformula, it is shown that the martin-gale arising in this way from Rf , where R is a Riesz transform in R n , isa martingale transform of the martingale arising from f . The two forma pair of martingales that have differential subordination and (in case ofHilbert or Riesz transforms) orthogonality. One then derives sharp martin-gale inequalities under hypotheses of strong differential subordination (andorthogonality) relations.In the case of Riesz transforms of the second order, the use of heat exten-sions in the upper half space instead of Poisson extensions originated in thecontext of a weighted estimate in Petermichl – Volberg [44] and was usedto prove L p estimates for the second order Riesz transforms based on theresults of Burkholder in Nazarov – Volberg [50] as part of their best-at-time estimate for the Beurling–Ahlfors operator, whose real and imaginaryparts themselves are second order Riesz transforms. We mention the re-cent version on discrete abelian groups
Domelevo–Petermichl [24] alsousing a type of heat flow. These proofs are deterministic. The techniqueof Bellman functions was used. This deterministic strategy does well whenno orthogonality is present and when strong subordination is the only im-portant property. Stochastic proofs (aside from giving better estimates insome situations) also have the advantage that once the integral representa-tion is known, the proofs are a very concise consequence of the respectivestatements on martingales.In [3] the authors proved sharp L p estimates for semi–discrete second orderRiesz transforms R α using stochastic integrals. There is an array of Riesztransforms of the second order that are treated, indexed my a matrix index α (see below for precisions on acceptable α ). The following representationformula of semi-discrete second order Riesz transforms R α `a la Gundy–Varopoulos (see [32]) is instrumental:
Theorem. (Arcozzi–Domelevo–Petermichl, 2016)
The second orderRiesz transform R α f of a function f ∈ L ( G ) as defined in (1.1) can bewritten as the conditional expectation E ( M α,f |Z = z ) . Here M α,ft is a suitable martingale transform of a martingale M ft associatedto f , and Z t is a suitable random walk on G EMI-DISCRETE RIESZ TRANSFORMS OF THE SECOND ORDER 3
We remark that the L p estimates of the discrete Hilbert transform on theintegers are still open. It is a famous conjecture that this operator has thesame norm as its continuous counterpart.These known L p norm inequalities use special functions found in the re-sults of Pichorides [45],
Verbitsky [18],
Ess´en [28],
Banuelos – Wang [12] when orthogonality is present in addition to differential subordinationor
Burkholder [15][16][17],
Wang [51] when differential subordination isthe only hypothesis.The aim of the present paper is to establish new estimates for semi–discrete Riesz transforms by using the martingale representation above to-gether with recent martingale inequalities found in the literature.Here is a brief description of the new results in this paper. • In the case where the function f is real valued, we can obtain betterestimates for R α than in the general case. These estimates dependupon the make of the matrix index α . The precise statement is foundin Theorem 1.2. • We prove a refined sharp weak type estimate using a weak type normdefined just before the statement of Theorem 1.3 . • We prove logarithmic and exponential estimates, in a sense limiting(in p ) cases of the classical sharp L p estimate. See Theorem 1.4. • We consider the norm estimates of the R α : L q → L p , spaces ofdifferent exponent. The statement is found in Theorem 1.5.1.1. Differential operators and Riesz transforms.First order derivatives and tangent planes.
We will consider Liegroups G := G x × G y , where G x is a discrete abelian group with a fixedset G of m generators, and their reciprocals, and G y is a connected, Liegroup of dimension n endowed with a biinvariant metric. The choice of theset G of generators in G x corresponds to the choice of a bi-invariant metricstructure on G x . We will use on G x the multiplicative notation for the groupoperation. We will define a product metric structure on G , which agreeswith the Riemannian structure on the first factor, and with the discrete“word distance” on the second. We will at the same time define a “tangentspace” T z G for G at a point z = ( x, y ) ∈ ( G x × G y ) = G . We will do thisin three steps.First, since G y is an n -dimensional connected Lie group with Lie alge-bra G y . We can identify each left-invariant vector field Y in G y with itsvalue at the identity e , G y ≡ T e G y . Since G is compact, it admits a bi-invariant Riemannian metric, which is unique up to a multiplicative factor.We normalize it so that the measure µ y associated with the metric satisfies µ y ( G y ) = 1. The measure µ y is also the normalized Haar measure of thegroup. We denote by < · , · > y be the corresponding inner product on T y G y and by ∇ y f ( y ) the gradient at y ∈ G y of a smooth function f : G y → R . K. DOMELEVO, A. OSE¸ KOWSKI, AND S. PETERMICHL
Let Y , . . . , Y n be a orthonormal basis for G y . The gradient of f can bewritten ∇ y f = Y ( f ) Y + . . . + Y n ( f ) Y n .Second, in the discrete component G x , let G x = ( g i ) i =1 ,...,m be a setof generators for G x , such that for i = j and σ = ± g i = g σj .The choice of a particular set of generators induces a word metric, hence, ageometry, on G x . Any two sets of generators induce bi-Lipschitz equivalentmetrics.At any point x ∈ G x , and given a direction i ∈ { , . . . , m } , we can definethe right and the left derivative at x in the direction i :( ∂ + f /∂x i )( x, y ) := f ( x + g i , y ) − f ( x, y ) := ( ∂ + i f )( x, y )( ∂ − f /∂x i )( x, y ) := f ( x, y ) − f ( x − g i , y ) := ( ∂ − i f )( x, y ) . Comparing with the continuous component, this suggests that the tangentplane ˆ T x G x at a point x of the discrete group G x might actually be split intoa “right” tangent plane T + x G x and a “left” tangent plane T − x G x , accordingto the direction with respect to which discrete differences are computed.We consequently define the augmented discrete gradient b ∇ x f ( x ), with a hat , as the 2 m –vector of ˆ T x G x := T + x G x ⊕ T − x G x accounting for all thelocal variations of the function f in the direct vicinity of x ; that is, the2 m –column–vector b ∇ x f ( x ) := ( X +1 f, X +2 f, . . . , X − f, X − f, . . . )( x ) = m X i =1 X τ = ± X τi f ( x )with X τi ∈ ˆ T x G x , where we noted the discrete derivatives X ± i f := ∂ ± i f andintroduced the discrete 2 m –vectors X ± i as the column vectors of Z m X + i = (0 , . . . , , . . . , × m , X − i = m × (0 , . . . , , . . . , . Here the 1’s in X ± i are located at the i –th position of respectively the firstor the second m –tuple. Notice that those vectors are independent of thepoint x . The scalar product on ˆ T x G x := T + x G x ⊕ T − x G x is defined as( U, V ) ˆ T x G x := 12 m X i =1 X τ = ± U τi V τi . We chose to put a factor in front of the scalar product to compensate forthe fact that we consider both left and right differences.Finally, for a function f defined on the cartesian product G := G x × G y ,the (augmented) gradient b ∇ z f ( z ) at the point z = ( x, y ) is an element ofthe tangent plane ˆ T z G := ˆ T x G x ⊕ T y G y , that is a (2 m + n )–column–vector b ∇ z f ( z ) := m X i =1 X τ = ± X τi f ( z ) ˆ X τi + n X j =1 Y j f ( z ) ˆ Y j ( z )= ( X +1 f, X +2 f, . . . , X − f, X − f, . . . , Y f, Y f, . . . )( z ) EMI-DISCRETE RIESZ TRANSFORMS OF THE SECOND ORDER 5 where ˆ X τi and ˆ Y j ( z ) can be identified with column vectors of size (2 m + n )with obvious definitions and scalar product ( · , · ) ˆ T z G z .Let d µ z := d µ x d µ y , d µ x being the counting measure on G x and d µ y beingthe Haar measure on G y . The inner product of ϕ, ψ in L ( G ) is( ϕ, ψ ) L ( G ) := Z G ϕ ( z ) ψ ( z ) dµ z ( z ) . Finally, we make the following hypotheses
Hypothesis.
We assume everywhere in the sequel:1. The discrete component G x of the Lie group G is an abelian group2. The connected component G y of the Lie group G is a Lie group thatcan be endowed with a biinvariant Riemannian metric, so that the family( Y j ) j =1 ,...,n commutes with ∆ y .Notice that this includes compact Lie groups G y since those can be en-dowed with a biinvariant metric. It also includes the usual Euclidian spacessince those are commutative. Riesz transforms.
Following [1][2], recall first that for a compact Rie-mannian manifold M without boundary, one denotes by ∇ M , div M and∆ M := div M ∇ M respectively the gradient, the divergence and the Lapla-cian associated with M . Then − ∆ M is a positive operator and the vectorRiesz transform is defined as the linear operator R M := ∇ M ◦ ( − ∆ M ) − / acting on L ( M ) ( L functions with vanishing mean). It follows that if f isa function defined on M and y ∈ M then R M f ( y ) is a vector of the tangentplane T y M .Similarly on M = G , we define ∇ G := b ∇ z as before, and then we definethe divergence operator as its formal adjoint, that is − div G = − c div z := b ∇ ∗ z ,with respect to the natural L inner product of vector fields:( U, V ) L ( ˆ T G ) := Z G ( U ( z ) , V ( z )) ˆ T z G d µ z ( z )We have the L -adjoints ( X ± i ) ∗ = − X ∓ i and Y ∗ j = − Y j . If U ∈ ˆ T G isdefined by U ( z ) = m X i =1 X τ = ± U τi ( z ) ˆ X τi + n X j =1 U j ( z ) ˆ Y j , we define its divergence b ∇ ∗ z U as b ∇ ∗ z U ( z ) := − m X i =1 X τ = ± X − τi U τi ( z ) − n X j =1 Y j U j ( z ) . K. DOMELEVO, A. OSE¸ KOWSKI, AND S. PETERMICHL
The Laplacian ∆ G is as one might expect:∆ z f ( z ) := − b ∇ ∗ z b ∇ z f ( z ) = − b ∇ ∗ x b ∇ x f ( z ) − b ∇ ∗ y b ∇ y f ( z )= m X i =1 X − i X + i f ( z ) + n X j =1 Y j f ( z )= m X i =1 X i f ( z ) + n X j =1 Y j f ( z )=: ∆ x f ( z ) + ∆ y f ( z )where we denoted X i := X + i X − i = X − i X + i . We have chosen signs so that − ∆ G > R z f )( z ) is the (2 m + n )–column–vector of the tangent plane ˆ T z G defined as the linear operatorˆ R z f := (cid:16) b ∇ z f (cid:17) ◦ ( − ∆ z f ) − / We also define transforms along the coordinate directions: R ± i = X ± i ◦ ( − ∆ z ) − / and R j = Y j ◦ ( − ∆ z ) − / . Plan of the paper.
In the next two sections, we present successivelythe main results of the paper and recall the weak formulations involvingsecond order Riesz transforms and semi-discrete heat extensions. Section 2introduces the stochastic setting for our problems. This includes in Subsec-tion 2.1 semi-discrete random walks, martingale transforms and quadraticcovariations. Subsection 2.2 presents a set of martingale inequalities alreadyknown in the literature. Finally, in Section 3 we give the proof of the mainresults.1.2.
Main results.
In this text, we are concerned with second order Riesztransforms and combinations thereof. We first define the square Riesz trans-form in the (discrete) direction i to be R i := R + i R − i = R − i R + i . Then, given α := (( α xi ) i =1 ...m , ( α yjk ) j,k =1 ...n ) ∈ C m × C n × n , we define R α tobe the following combination of second order Riesz transforms:(1.1) R α := m X i =1 α xi R i + n X j,k =1 α yjk R j R k , where the first sum involves squares of discrete Riesz transforms as definedabove, and the second sum involves products of continuous Riesz transforms.This combination is written in a condensed manner as the quadratic form R α = (cid:16) ˆ R z , A α ˆ R z (cid:17) EMI-DISCRETE RIESZ TRANSFORMS OF THE SECOND ORDER 7 where A α is the (2 m + n ) × (2 m + n ) block matrix(1.2) A α := (cid:18) A xα A yα (cid:19) with A xα = diag ( α x , . . . , α xm , α x , . . . , α xm ) ∈ C m × m , A yα = ( α yjk ) j,k =1 ...n ∈ C n × n . In the theorems below, we assume that G is a Lie group and R α is acombination of second order Riesz transforms as defined above. The firstapplication of the stochastic integral formula, Theorem 1.1 was done in [3],while the other applications, Theorems 1.2 1.3 1.4 and 1.5 are new. Theorem 1.1. (Arcozzi–Domelevo–Petermichl, 2016)
For any
Ose¸kowski [10].
Theorem 1.2.
Assume that a I A α b I in the sense of quadratic forms.Then R α : L p ( G , R ) → L p ( G , R ) enjoys the norm estimate k R α k p C a,b,p . We should point out here that the constants C a,b,p appear in earlier worksof Burkholder [15] (for a = − b : then C a,b,p = b ( p ∗ − Choi (in the case when one of a , b is zero). The Choi constants arenot explicit; an approximation of C , ,p is known and writes as C , ,p = p + log (cid:16) e − (cid:17) + β p + . . . ., with β = log (cid:16) e − (cid:17) + log (cid:16) e − (cid:17) − (cid:16) e − e − (cid:17) . Coming back to complex-valued functions, we will also establish the fol-lowing weak-type bounds. We consider the norms || f || L p, ∞ ( G , C ) = sup (cid:26) µ z ( E ) /p − Z E f dµ z (cid:27) , where the supremum is taken over the class of all measurable subsets E of G of positive measure. K. DOMELEVO, A. OSE¸ KOWSKI, AND S. PETERMICHL
Theorem 1.3.
For any < p < ∞ we have k R α k L p ( G , C ) → L p, ∞ ( G , C ) k A α k · (cid:18)
12 Γ (cid:18) p − p − (cid:19)(cid:19) − /p if < p ≤ , (cid:18) p p − (cid:19) /p if p ≥ . We will also prove the following logarithmic and exponential estimates,which can be regarded as versions of Theorem 1.1 for p = 1 and p = ∞ .Consider the Young functions Φ , Ψ : [0 , ∞ ) → [0 , ∞ ), given by Φ( t ) = e t − − t and Ψ( t ) = ( t + 1) log( t + 1) − t. Theorem 1.4.
Let
K > be fixed.(i) For any measurable subset E of G and any f on G we have Z E | R α f | dµ z k A α k · (cid:18) K Z G Ψ( | f | ) dµ z + µ z ( E )2( K − (cid:19) . (ii) For any f : G → C bounded by , Z G Φ (cid:18) | R α f | K k A α k (cid:19) dµ z || f || L ( G , C ) K ( K − . Our final result concerns another extension of Theorem 1.1, which studiesthe action of R α between two different L p spaces. For 1 ≤ p < q < ∞ , let C p,q be the constant defined by Ose¸kowski in [40].
Theorem 1.5.
For any ≤ p < q < ∞ , any measurable subset E of G andany f ∈ L q ( G ) we have k R α f k L p ( E, C ) C p,q k A α k k f k L q ( G , C ) µ z ( E ) /p − /q . An interesting feature is that all the estimates in the five theorems aboveare sharp when the group G = G x × G y and dim( G y ) + dim ∞ ( G x ) > ∞ ( G x ) denotes the number of infinite components of G x .1.3. Weak formulations.
Let f : G → C be given. The heat extension˜ f ( t ) of f is defined as ˜ f ( t ) := e t ∆ z f =: P t f . We have therefore ˜ f (0) = f .The aim of this section is to derive weak formulations for second order Riesztransforms. We start with the weak formulation of the identity operator I ,that is obtained by using semi-discrete heat extensions (see [3] for details). EMI-DISCRETE RIESZ TRANSFORMS OF THE SECOND ORDER 9
Assume f in L ( G ) and g in L ( G ). Let ¯ f be the average of f on G if G has finite measure and zero otherwise. Then( I f, g ) = ( f, g ) L ( G ) = ¯ f ¯ g + 2 Z ∞ (cid:16) b ∇ z P t f, b ∇ z P t g (cid:17) L ( ˆ T G ) d t = ¯ f ¯ g + 2 Z ∞ Z z ∈ G ( m X i =1 X τ = ± ( X τi P t f )( z )( X τi P t g )( z ) + n X j =1 ( Y j P t f )( z )( Y j P t g )( z ) d µ z ( z )d t and the sums and integrals that arise converge absolutely.In order to pass to the weak formulation for the squares of Riesz trans-forms, we first observe that the following commutation relations hold Y j ◦ ∆ z = ∆ z ◦ Y j X τi ◦ ∆ z = ∆ z ◦ X τi , τ ∈ { + , −} This is an easy consequence of the hypothesis made on the Lie group. Fol-lowing [3], the following weak formulation for second order Riesz transformsholdsAssume f in L ( G ) and g in L ( G ), then( R α f, g ) L ( G ) = − Z ∞ (cid:16) A α b ∇ z P t f, b ∇ z P t g (cid:17) L ( ˆ T G ) d t = − Z ∞ Z z ∈ G ( m X i =1 X τ = ± α xi ( X τi P t f )( z )( X τi P t g )( z )+ n X j,k =1 α yjk ( Y j P t f )( z )( Y k P t g )( z ) ) d µ z ( z ) d t and the sums and integrals that arise converge absolutely.2. Stochastic integrals and martingale transforms
In what follows, we assume that we have a complete probability space(Ω , F , P ) with a c`adl`ag (i.e. right continuous left limit) filtration ( F t ) t > of sub- σ –algebras of F . We assume as usual that F contains all eventsof probability zero. All random walks and martingales are adapted to thisfiltration.We define below a continuous-time random process Z with values in G , Z t := ( X t , Y t ) ∈ G x × G y , having infinitesimal generator L = ∆ z . The pure-jump component X t is a compound Poisson jump process on the discrete set G x , wheras the continuous component Y t is a standard brownian motion onthe manifold G y . Then, Itˆo’s formula ensures that semi-discrete “harmonic”functions f : R + × G → C solving the backward heat equation ( ∂ t + ∆ z ) f = M ft := f ( t, Z t ) for which we define a class ofmartingale transforms.2.1. Stochastic integrals, Martingale transforms and quadratic co-variations.Stochastic integrals on Riemannian manifolds and Itˆo integral.
Following
Emery [26][27], see also
Arcozzi [1][2], we define the Brownianmotion Y t on G y , a compact Riemannian manifold, as the process Y t : Ω → (0 , T ) × G y such that for all smooth functions f : G y → R , the quantity(2.1) f ( Y t ) − f ( Y ) − Z t (∆ y f )( Y s ) d s =: ( I d y f ) t is an R –valued continuous martingale. For any adapted continuous processΨ with values in the cotangent space T ∗ G y of G y , if Ψ t ( ω ) ∈ T ∗ Y t ( ω ) G y forall t > ω ∈ Ω, then one can define the continuous
Itˆo integral I Ψ ofΨ as ( I Ψ ) t := Z t h Ψ s , d Y s i so that in particular ( I d y f ) t := Z t h d y f ( Y s ) , d Y s i The integrand above involves the 1–form of T ∗ y G y d y f ( y ) := X j ( Y j f )( y ) Y ∗ j . A pure jump process on G x . We will now define the discrete m –dimensional process X t on the discrete abelian group G x as a generalizedcompound Poisson process. In order to do this we need a number of inde-pendent variables and processes:First, for any given 1 i m , let N it be a c`adl`ag Poisson process ofparameter λ , that is ∀ t, P ( N it = n ) = ( λt ) n n ! e − λt . The sequence of instants where the jumps of the N it occur is noted ( T ik ) k ∈ N ,with the convention T i = 0.Second, we set N t = m X i =1 N it Almost surely, for any two distinct i and j , we have { T ik } k ∈ N ∩ { T jk } k ∈ N = ∅ .Let therefore { T k } k ∈ N = ∪ mi =1 { T ik } k ∈ N be the ordered sequence of instants ofjumps of N t and let i t ≡ i t ( ω ) be the index of the coordinate where the jump EMI-DISCRETE RIESZ TRANSFORMS OF THE SECOND ORDER 11 occurs at time t . We set i t = 0 if no jump occurs. The random variables i t are measurable: i t = ( N t − N t − , N t − N t − , . . . , N mt − N mt − ) · (1 , , . . . , m ).In differential form, d N t = m X i =1 d N it = d N i t t . Third, we denote by ( τ k ) k ∈ N a sequence of independent Bernoulli variables ∀ k, P ( τ k = 1) = P ( τ k = −
1) = 1 / . Finally, the random walk X t started at X ∈ G x is the c`adl`ag compoundPoisson process (see e.g. Protter [48],
Privault [46, 47]) defined as X t := X + N t X k =1 G τ k i k , where G τi = (0 , . . . , , τ g i , , . . . ,
0) when i = 0 and (0 , . . . ,
0) when i = 0. Stochastic integrals on discrete groups.
We recall for the convenienceof the reader the derivation of stochastic integrals for jump processes. Wewill emphasize the fact that the corresponding Itˆo’s formula involves theaction of a discrete 1–form written in a well-chosen local coordinate system ofthe discrete augmented cotangent plane (see details below). Let 1 k N t and let ( T k , i k , τ k ) be respectively the instant, the axis and the direction ofthe k –th jump. We set T = 0. Let f := f ( t, x ), t ∈ R + , x ∈ G x a functiondefined on R + × G x . Then f ( t, X t ) − f (0 , X )= Z t ( ∂ t f )( s, X s )d s + m X i =1 Z t ( f ( s, X s ) − f ( s, X s − )) d N is . At an instant s , the integrand in the last term writes as( f ( s, X s ) − f ( s, X s − ))d N is = (cid:0) f (cid:0) s, X s − + G τ N s i (cid:1) − f ( s, X s − ) (cid:1) d N is = (cid:0) X τ N s i f (cid:1) ( s, X s − ) τ N s d N is = 12 (cid:8) ( X i f )( s, X s − ) + τ N s ( X i f )( s, X s − ) (cid:9) d N is where we introduced, for all 1 i m , X i := X + i + X − i X i := X + i − X − i . Notice that, for any given 1 i m , up to a normalisation factor, thesystem of coordinate ( X i , X i ) is obtained thanks to a rotation of π/ canonical system of coordinate ( X + i , X − i ). Finally, f ( t, X t ) − f (0 , X )= Z t (cid:26) ( ∂ t f )( s, X s ) + λ x f )( s, X s ) (cid:27) d s + Z t Db d f ( s, X s − ) , d c W s E =: Z t (cid:26) ( ∂ t f )( s, X s ) + λ x f )( s, X s ) (cid:27) d s + (cid:16) I b d x f (cid:17) t . where we set d X is = τ N s d N is . It is easy to see that d X is is the stochasticdifferential of a martingale. Here and in the sequel, we take λ = 2. Discrete Itˆo integral.
The stochastic integral above shows that Itˆo for-mula (2.1) for continuous processes has a discrete counterpart involving sto-chastic integrals for jump processes, namely we have the discrete
Itˆo integral (cid:16) I b d x f (cid:17) t := 12 m X i =1 Z t ( X i f )( s, X s − ) d( N is − λs ) + ( X i f )( s, X s − ) d X s This has a more intrinsic expression similar to the continuous Itˆo integral(2.1). If we regard the discrete component G x as a “discrete Riemannian”manifold, then this discrete Itˆo integral involves discrete vectors (resp. 1–forms) defined on the augmented discrete tangent (resp. cotangent) spaceˆ T x G x (resp. ˆ T ∗ x G x ) of dimension 2 m defined asˆ T x G x = span { X +1 , X +2 , . . . , X − , X − , . . . } = span { X , X , . . . , X , X , . . . } ˆ T ∗ x G x = span { ( X +1 ) ∗ , ( X +2 ) ∗ , . . . , ( X − ) ∗ , ( X − ) ∗ , . . . } = span { ( X ) ∗ , ( X ) ∗ , . . . , ( X ) ∗ , ( X ) ∗ , . . . } . Let d c W s ∈ ˆ T X s G x be the vector and b d f ∈ ˆ T ∗X s G x be the 1–form respectivelydefined as:d c W s = d( N s − λs ) X + . . . + d( N ms − λs ) X m + d X s X + . . . + d X ms X m b d x f = X f ( X ) ∗ + . . . + X m f ( X m ) ∗ + X f ( X ) ∗ + . . . + X m f ( X m ) ∗ We have with these notations (cid:16) I b d x f (cid:17) t := Db d x f, d c W s E ˆ T ∗ x G x × ˆ T x G x where the factor 1 / h· , ·i ˆ T ∗ x G x × ˆ T x G x . Semi–discrete stochastic integrals.
Let finally Z t = ( X t , Y t ) be a semi-discrete random walk on the cartesian product G = G x × G y , where X t isthe random walk above defined on G x with generator ∆ x and where Y t isthe Brownian motion defined on G y with generator ∆ y . For f := f ( t, z ) = EMI-DISCRETE RIESZ TRANSFORMS OF THE SECOND ORDER 13 f ( t, x, y ) defined from R + × G onto C , we have easily the stochastic integralinvolving both discrete and continuous parts: f ( t, Z t ) = Z t { ( ∂ t f )( s, Z s ) + (∆ z f )( s, Z s ) } d s + (cid:16) I b d z f (cid:17) t where the semi-discrete Itˆo integral writes as (cid:16) I b d z f (cid:17) t := (cid:16) I b d x f (cid:17) t + (cid:0) I d y f (cid:1) t := Z t Db d x f ( s, Z s − ) , d c W s E ˆ T ∗X s G x × ˆ T X s G x + Z t (cid:10) d y f ( s, Z s − ) , d Y s (cid:11) ˆ T ∗Y s G y × ˆ T Y s G y . Martingale transforms.
We are interested in martingale transformsallowing us to represent second order Riesz transforms. Let f ( t, z ) be asolution to the heat equation ∂ t − ∆ z = 0. Fix T > Z ∈ G . Thendefine ∀ t T, M f,T, Z t = f ( T − t, Z t ) . This is a martingale since f ( T − t ) solves the backward heat equation ∂ t +∆ z = 0, and we have in terms of stochastic integrals M f,T, Z t = f ( T − t, Z t ) = f ( T, Z ) + Z t Db d z f ( T − s, Z s − ) , d Z s E Given A α the C (2 m + n ) × (2 m + n ) matrix defined earlier, we note M α,f,T, Z t themartingale transform A α ∗ M f,T, Z t defined as M α,f,T, Z t := f ( T, Z ) + Z t (cid:16) A α b ∇ z f ( s, Z s − ) , d Z s (cid:17) = f ( T, Z ) + Z t Db d z f ( T − s, Z s − ) A ∗ α , d Z s E where the first integral involves the L scalar product on ˆ T z G × ˆ T z G andthe second integral involves the duality ˆ T ∗ z G × ˆ T z G . In differential form:d M α,f,T, Z t = (cid:16) A α b ∇ z f ( s, Z s − ) , d Z s (cid:17) = m X i =1 α xi (cid:8) ( X i f )( T − t, Z t − ) d( N it − λt ) + ( X i f )( t, Z t − ) d X it (cid:9) + n X j =1 α yj,k ( X j f )( T − t, Z t − ) d Y kt Quadratic covariation and subordination.
We have the quadratic co-variations (see
Protter [48],
Dellacherie – Meyer [22], or
Privault [46, 47]). Since d[ N i − λt, N i − λt ] t = d N it d[ N i − λt, X i ] t = τ N t d N it d[ X i , X i ] t = d N it d[ Y j , Y j ] t = d t, it follows thatd[ M f , M g ] t = m X i =1 X τ = ± ( X τi f ) ( X τi g )( T − t, Z t − ) ( τ N t = τ N it (2.2) + ( ∇ y f, ∇ y g ) ( T − t, Z t − )d t Differential subordination.
Following
Wang [51], given two adaptedc`adl`ag Hilbert space valued martingales X t and Y t , we say that Y t is dif-ferentially subordinate by quadratic variation to X t if | Y | H | X | H and[ Y, Y ] t − [ X, X ] t is nondecreasing nonnegative for all t . In our case, we haved[ M α,f , M α,f ] t = m X i =1 | α xi | (cid:8) ( X + i f ) ( T − t, Z t − ) ( τ N t = 1)+ ( X − i f ) ( T − t, Z t − ) ( τ N t = − (cid:9) d N it + ( A yα ∇ y f, A yα ∇ y f ) ( T − t, Z t − )d t. Hence(2.3) d[ M α,f , M α,f ] t k A α k d[ M f , M f ] t . This means that M α,ft is differentially subordinate to k A α k M ft .2.2. Martingale inequalities under differential subordination.
In thefinal part of the section we discuss a number of sharp martingale inequalitieswhich hold under the assumption of the differential subordination imposedon the processes. Our starting point is the following celebrated L p bound. Theorem 2.1. (Wang, 1995)
Suppose that X and Y are martingalestaking values in a Hilbert space H such that Y is differentially subordinateto X . Then for any < p < ∞ we have || Y || p ≤ ( p ∗ − || X || p and the constant p ∗ − is the best possible, even if H = R . This result was first proved by
Burkholder in [15] in the followingdiscrete-time setting. Suppose that ( X n ) n ≥ is an H -valued martingale and EMI-DISCRETE RIESZ TRANSFORMS OF THE SECOND ORDER 15 ( α n ) n ≥ is a predictable sequence with values in [ − , Y := α ∗ X bethe martingale transform of X defined for almost all ω ∈ Ω by Y ( ω ) = α X ( ω ) and ( Y n +1 − Y n )( ω ) = α n ( X n +1 − X n )( ω ) . Then the above L p bound holds true and the constant p ∗ − Wang [51]. Tosee that the preceding discrete-time version is indeed a special case, treat adiscrete-time martingale ( X n ) n ≥ and its transform ( Y n ) n ≥ as continuous-time processes via X t = X ⌊ t ⌋ , Y t = Y ⌊ t ⌋ for t ≥
0; then Y is differentiallysubordinate to X .In 1992, Choi [19] established the following non-symmetric, discrete-timeversion of the L p estimate. Theorem 2.2. (Choi, 1992)
Suppose that ( X n ) n ≥ is a real-valued dis-crete time martingale and let ( Y n ) n ≥ be its transform by a predictable se-quence ( α n ) n ≥ taking values in [0 , . Then there exists a constant C p de-pending only on p such that k Y k p C p k X k p and the estimate is best possible. This result can be regarded as a non-symmetric version of the previoustheorem, since the transforming sequence ( α n ) n ≥ takes values in a non-symmetric interval [0 , Ba˜nuelos and
Ose¸kowski addresses both these questions. For any real numbers a < b and any 1 < p < ∞ , let C a,b,p be the constant introduced in [10]. Theorem 2.3. (Banuelos–Os¸ekowski, 2012)
Let ( X t ) t ≥ and ( Y t ) t ≥ be two real-valued martingales satisfying (2.4) d (cid:20) Y − a + b X, Y − a + b X (cid:21) t d (cid:20) b − a X, b − a X (cid:21) t for all t > . Then for all < p < ∞ , we have k Y k p C a,b,p k X k p . The condition (2.4) is the continuous counterpart of the condition thatthe transforming sequence ( α n ) n ≥ takes values in the interval [ a, b ]. Thus,in particular, Choi’s constant C p is, in the terminology of the above theorem,equal to C p, , .We return to the context of the “classical” differential subordination in-troduced in the preceding subsection and study other types of martingale in-equalities. The following statements, obtained by Ba˜nuelos – Ose¸kowski ,[11] will allow us to deduce sharp weak-type and logarithmic estimates forRiesz transforms, respectively.
Theorem 2.4. (Banuelos–Os¸ekowski, 2015)
Suppose that X and Y aremartingales taking values in a Hilbert space H such that Y is differentiallysubordinate to X . (i) Let < p < . Then for any t ≥ , E max ( | Y t | − p − / ( p − (cid:18) pp − (cid:19) , ) ≤ E | X t | p . (ii) Suppose that < p < ∞ . Then for any t ≥ , E max (cid:8) | Y t | − p − , (cid:9) ≤ p p − E | X t | p . Both estimates are sharp: for each p , the numbers p − / ( p − Γ (cid:16) pp − (cid:17) and − p − cannot be decreased. Recall that Φ , Ψ : [0 , ∞ ) → [0 , ∞ ) are conjugate Young functions givenby Φ( t ) = e t − − t and Ψ( t ) = ( t + 1) log( t + 1) − t . Theorem 2.5. (Banuelos–Os¸ekowski, 2015)
Suppose that X and Y aremartingales taking values in a Hilbert space H such that Y is differentiallysubordinate to X . Then for any K > and any t ≥ we have E (cid:8) | Y t | − (2( K − − , (cid:9) ≤ K E Ψ( | X t | ) . For each K , the constant (2( K − − appearing on the left, is the bestpossible (it cannot be replaced by any smaller number). The following exponential estimate, established by
Ose¸kowski in [41],can be regarded as a dual statement to the above logarithmic bound.
Theorem 2.6. (Os¸ekowski, 2013)
Assume that X , Y are H -valued mar-tingales such that || X || ∞ ≤ and Y is differentially subordinate to X . Thenfor any K > and any t ≥ we have (2.5) E Φ( | Y t | /K ) ≤ K ( K − E | X t | . Finally, we will need the following sharp L q → L p estimate, establishedby Ose¸kowski in [43], which will allow us to deduce the correspondingestimate for Riesz transforms.
Theorem 2.7. (Os¸ekowski, 2014)
Assume that X , Y are H -valuedmartingales such that Y is differentially subordinate to X . Then for any ≤ p < q < ∞ there is a constant L p,q such that (2.6) E max {| Y t | p − L p,q , } ≤ E | X t | q . Actually, the paper [43] identifies, for any p and q as above, the optimal(i.e., the least) value of the constant L p,q in the estimate above. As thedescription of this constant is a little complicated (and will not be needed inour considerations below), we refer the reader to that paper for the formaldefinition of L p,q .Let us conclude with the observation which will be crucial in the proofsof our main results. Namely, all the martingale inequalities presented aboveare of the form E ζ ( | Y t | ) ≤ E ξ ( | X t | ), t ≥
0, where ζ , ξ are certain convex EMI-DISCRETE RIESZ TRANSFORMS OF THE SECOND ORDER 17 functions. This will allow us to successfully apply a conditional version ofJensen’s inequality. 3.
Proofs of the main results
We turn our attention to the proofs of the estimates for R α formulated inthe introductory section. We will focus on Theorems 1.1, 1.2 and 1.3 only;the remaining statements are established by similar arguments. Also, wepostpone the proof of the sharpness of these estimates to the next section.3.1. Proof of Theorem 1.1.
Recall that the subordination estimate (2.3)shows that the martingale transform Y t := M αt is differentially subordinateto the martingale X t := k A α k M ft . Therefore, by Theorem 2.1, we imme-diately obtain that k M α,ft k p k A α k ( p ∗ − k M ft k p for all t ≥
0. Since the operator T α is a conditional expectation of M α,ft , anapplication of Jensen’s inequality proves the estimate kT α k p k A α k ( p ∗ − Proof of Theorem 1.2.
The argument is the same as above andexploits the fine-tuned L p estimate of Theorem 2.3 applied to X t = M ft and Y t = M α,ft . It is not difficult to prove that the difference of quadraticvariations above writes in terms of a jump part and a continuous part as (cid:20) Y − a + b X, Y − a + b X (cid:21) t − d (cid:20) b − a X, b − a X (cid:21) t = m X i =1 X ± ( α xi − a )( α xi − b ) ( X ± i f ) ( B t ) ( τ N t = ±
1) d N it + h ( A yα − a I ) ( A yα − b I ) ∇ y f ( B t ) , ∇ y f ( B t ) i d t, which is nonpositive since we assumed precisely a I A α b I . Thus, theestimate of Theorem 1.2 follows. The sharpness is established in a similarmanner. (cid:3) Proof of Theorem 1.3.
We will focus on the case 1 < p <
2; forremaining values of p the argument is similar. An application of Theorem2.4 to the processes X t = k A α k M ft and Y t = M α,ft yields E max ( | M α,ft | − p − / ( p − (cid:18) pp − (cid:19) , ) ≤ k A α k p E | M ft | p and hence, by Jensen’s inequality, we obtain Z G max ( | R α f | − p − / ( p − (cid:18) pp − (cid:19) , ) dµ z ≤ k A α k p || f || pL p ( G ) . Therefore, if E is an arbitrary measurable subset of G , we get Z E | R α f | dµ z ≤ Z E | R α f | − p − / ( p − (cid:18) pp − (cid:19)! d µ z + p − / ( p − (cid:18) pp − (cid:19) µ z ( E ) ≤ || f || pL p ( G ) + p − / ( p − (cid:18) pp − (cid:19) µ z ( E ) . Apply this bound to λf , where λ is a nonnegative parameter, then divideboth sides by λ and optimize the right-hand side over λ to get the desiredassertion. 4. Sharpness
The proof of the sharpness of the different results is made in several steps.In some cases the sharpness for certain second order Riesz transform esti-mates in the continuous setting (such as in Theorem 1.1) is already known.In these cases we prove below the sharpness for the discrete (or semidiscrete)case by using sequences of finite difference approximates of continuous func-tions and their finite difference second order Riesz transforms. In othercases, we need to prove first sharpness for certain continuous second orderRiesz transforms. The key point here is to transfer the sharp result forzigzag martingales into a sharp result for certain continuous second orderRiesz transforms by the laminate technique. We will illustrate this for theweak-type estimate of Theorem 1.3 and establish the following statement.
Theorem 4.1.
Let
Θ : [0 , ∞ ) → [0 , ∞ ) be a given function and let λ > be afixed number. Assume further that there is a pair ( F, G ) of finite martingalesstarting from (0 , such that G is a ± -transform of F and E ( | G ∞ | − λ ) + > E Θ( | F ∞ | ) . Then there is a function f : R → R supported on the unit disc D of R suchthat Z R (cid:0) | ( R − R ) f | − λ (cid:1) + d x > Z D Θ( | f | ) d x. We will prove this statement with the use of laminates, important familyof probability measures on matrices. It is convenient to split this sectioninto several separate parts. For the sake of convenience, and to make thissection as self contained as possible, we recall the preliminaries on laminatesand their connections to martingales from [14] and [39], Section 4.2.4.1.
Laminates.
Assume that R m × n stands for the space of all real matricesof dimension m × n and R n × nsym denote the subclass of R n × n which consistsof all symmetric matrices of dimension n × n . EMI-DISCRETE RIESZ TRANSFORMS OF THE SECOND ORDER 19
Definition 4.2.
A function f : R m × n → R is said to be rank-one convex , iffor all A, B ∈ R m × n with rank B = 1, the function t f ( A + tB ) is convexFor other equivalent definitions of rank-one convexity, see [21, p. 100].Suppose that P = P ( R m × n ) is the class of all compactly supported proba-bility measures on R m × n . For a measure ν ∈ P , we define ν = Z R m × n Xdν ( X ) , the associated center of mass or barycenter of ν. Definition 4.3.
We say that a measure ν ∈ P is a laminate (and write ν ∈ L ), if f ( ν ) ≤ Z R m × n f d ν for all rank-one convex functions f . The set of laminates with barycenter 0is denoted by L ( R m × n ).Laminates can be used to obtain lower bounds for solutions of certainPDEs, as observed by Faraco in [30]. In addition, laminates appear nat-urally in the context of convex integration, where they lead to interestingcounterexamples, see e.g. [5], [20], [34], [37] and [49]. For our results here wewill be interested in the case of 2 × Definition 4.4.
Let U be a subset of R × and let PL ( U ) denote thesmallest class of probability measures on U which(i) contains all measures of the form λδ A + (1 − λ ) δ B with λ ∈ [0 ,
1] andsatisfying rank( A − B ) = 1;(ii) is closed under splitting in the following sense: if λδ A + (1 − λ ) ν belongs to PL ( U ) for some ν ∈ P ( R × ) and µ also belongs to PL ( U ) with µ = A , then also λµ + (1 − λ ) ν belongs to PL ( U ).The class PL ( U ) is called the prelaminates in U .It follows immediately from the definition that the class PL ( U ) only con-tains atomic measures. Also, by a successive application of Jensen’s in-equality, we have the inclusion PL ⊂ L . The following are two well knownlemmas in the theory of laminates; see [5], [33], [37], [49].
Lemma 4.5.
Let ν = P Ni =1 λ i δ A i ∈ PL ( R × sym ) with ν = 0 . Moreover, let < r < min | A i − A j | and δ > . For any bounded domain B ⊂ R thereexists u ∈ W , ∞ ( B ) such that k u k C < δ and for all i = 1 . . . N (cid:13)(cid:13) { x ∈ B : | D u ( x ) − A i | < r } (cid:13)(cid:13) = λ i |B| . Lemma 4.6.
Let K ⊂ R × sym be a compact convex set and suppose that ν ∈ L ( R × sym ) satisfies supp ν ⊂ K . For any relatively open set U ⊂ R × sym with K ⊂ U , there exists a sequence ν j ∈ PL ( U ) of prelaminates with ν j = ν and ν j ∗ ⇀ ν , where ∗ ⇀ denotes weak convergence of measures. Combining these two lemmas and using a simple mollification, we obtainthe following statement, proved by Boros, Sh´ekelyhidi Jr. and Volberg [14].It exhibits the connection between laminates supported on symmetric ma-trices and second derivatives of functions. It will be our main tool in theproof of the sharpness. Recall that D denotes the unit disc of C . Corollary 4.7.
Let ν ∈ L ( R × sym ) . Then there exists a sequence u j ∈ C ∞ ( D ) with uniformly bounded second derivatives, such that | D | Z D φ ( D u j ( x )) d x → Z R × sym φ d ν for all continuous φ : R × sym → R . Biconvex functions and a special laminate.
The next step in ouranalysis is devoted to the introduction of a certain special laminate. Weneed some additional notation. A function ζ : R × R → R is said to be biconvex if for any fixed z ∈ R , the functions x ζ ( x, z ) and y ζ ( z, y )are convex. Now, take the martingales F and G appearing in the statementof Theorem 4.1. Then the martingale pair( F , G ) := (cid:18) F + G , F − G (cid:19) is finite, starts from (0 ,
0) and has the following zigzag property: for any n ≥ F n = F n +1 with probability 1 or G n = G n +1 almost surely;that is, in each step ( F , G ) moves either vertically, or horizontally. Indeed,this follows directly from the assumption that G is a ± F . Thisproperty combines nicely with biconvex functions: if ζ is such a function,then a successive application of Jensen’s inequality gives(4.1) E ζ ( F n , G n ) ≥ E ζ ( F n − , G n − ) ≥ . . . ≥ E ζ ( F , G ) = ζ (0 , . The distribution of the terminal variable ( F ∞ , G ∞ ) gives rise to a proba-bility measure ν on R × sym : put ν (diag( x, y )) = P (cid:0) ( F ∞ , G ∞ ) = ( x, y ) (cid:1) , ( x, y ) ∈ R , where diag( x, y ) stands for the diagonal matrix (cid:18) x y (cid:19) . Observe that ν is a laminate of barycenter 0. Indeed, if ψ : R × → R is a rank-one convex,then ( x, y ) ψ (diag( x, y )) is biconvex and thus, by (4.1), Z R × ψ d ν = E ψ (diag( F ∞ , G ∞ )) ≥ ψ (diag(0 , ψ (¯ ν ) . Here we used the fact that ( F , G ) is finite, so ( F ∞ , G ∞ ) = ( F n , G n ) for some n . EMI-DISCRETE RIESZ TRANSFORMS OF THE SECOND ORDER 21
A proof of Theorem 4.1.
Consider a continuous function φ : R × sym → R given by φ ( A ) = ( | A − A | − λ ) + − Θ( | A + A | ) . By Corollary 4.7, there is a functional sequence ( u j ) j ≥ ⊂ C ∞ ( D ) such that1 | D | Z R φ ( D u j )d x = 1 | D | Z D φ ( D u j )d x j →∞ −−−→ Z R × sym φ d ν = E ( | G ∞ | − λ ) + − E Θ( | F ∞ | ) > . Therefore, for sufficiently large j , we have Z R (cid:18)(cid:13)(cid:13)(cid:13)(cid:13) ∂ u j ∂x − ∂ u j ∂y (cid:13)(cid:13)(cid:13)(cid:13) − λ (cid:19) + d x d y > Z R Θ ( | ∆ u j | ) d x d y. Setting f = ∆ u j , we obtain the desired assertion.In the remaining part of this subsection, let us briefly explain how The-orem 4.1 yields the sharpness of weak-type and logarithmic estimates forsecond-order Riesz transforms (in the classical setting). We will focus onthe weak-type bounds for 1 < p < λ p is the best constant in the estimate E ( | G ∞ | − λ p ) + ≤ E | F ∞ | p , valid for all pairs ( F, G ) of finite martingales starting from 0 such that G isa ± F . The value of λ p appears in the statement of Theorem9 above, the fact that it is already the best for martingale transforms followsfrom the examples exhibited in [38]. For any ε >
0, Theorem 4.1 yields theexistence of f : R → R , supported on the unit disc, such that Z R (cid:0) | ( R − R ) f | − λ p + ε (cid:1) + d x d y > Z R | f | p d x d y. That is, if we set A = {| ( R − R ) f | ≥ λ p − ε } , we get(4.2) Z A | ( R − R ) f | d x d y > Z R | f | p d x d y + ( λ p − ε ) | A | . However, if the weak-type estimate holds with a constant c p , Young’s in-equality implies Z A | ( R − R ) f | d x d y ≤ Z R | f | p d x d y + ( p − c p/ ( p − p p p/ ( p − | A | . Therefore, the inequality (4.2) enforces that( p − c p/ ( p − p p p/ ( p − ≥ λ p (since ε was arbitrary). This estimate is equivalent to c p ≥ (cid:18)
12 Γ (cid:18) p − p − (cid:19)(cid:19) − /p , which is the desired sharpness.4.4. From continuous to discrete sharp estimates.
We claim that thesharp bounds found for the continuous second order Riesz transforms alsohold in the case of purely discrete groups. Groups of mixed type wouldbe treated in the same manner. We illustrate those results only for thesharpness in Theorem 1.1 and in Theorem 1.3 since other results followthe same lines. Precisely, we show that the sharpness in the discrete caseis inherited from the sharpness of the continuous case through the use ofthe so–called fundamental theorem of finite difference methods from Laxand Richtmyer [35] (see also [36]). This result states that stability and consistency of the finite difference scheme implies convergence of theapproximate finite difference solution towards the continuous solution, in asense that we detail below.
Finite difference Riesz transforms.
Let u = R i f be the i -th secondorder Riesz transform in Ω := R N of a function f ∈ L p . The function u isthe unique solution to the Poisson problem in R N , ∆ u = ∂ i f in R N (see[29]). This is a problem of the form Au = Bf , where A = ∆ and B = ∂ ij .Introduce now a finite difference grid of step–size h >
0, that is the gridΩ h := h Z N . The functions v h defined on Ω h are equipped with the L ph normdefined as k v h k pL ph := X x ∈ Ω h | v h ( x ) | p h N . It is common to identify a finite difference function v h defined on the gridΩ h with the piecewise constant function (also denoted) v h : R N → C suchthat v h ( x ) = v h ( y ) for all x ’s in the open cube Ω( y ) of volume h N centeredaround the grid point y ∈ Ω h . With this notation, we might write finitedifference integrals in the form k v h k L ph = Z x ∈ R N | v h ( x ) | d µ h ( x ) . The finite difference second order Riesz transform u h = R i f h of f h is thesolution to the problem A h u h = B h f h , where A h := ∆ h is the finite dif-ference Laplacian and B h := ∂ i,h the 3–point finite difference second orderderivative. Precisely, for any x ∈ Ω h , any v h : Ω h → R ,( ∂ i,h v h )( x ) := v h ( x + he i ) − v h ( x ) + v h ( x − he i ) h (∆ h v h )( x ) := N X i =1 ( ∂ i,h v h )( x ) . It is classical that we have the consistency of the discrete problem withrespect to the continuous problem, that is for given smooth functions u and f we have ∆ h u = ∆ u + O ( h ) and ∂ i,h f = ∂ i f + O ( h ), where thecoefficients in O ( h ) include as a factor up to fourth–order derivatives of u EMI-DISCRETE RIESZ TRANSFORMS OF THE SECOND ORDER 23 or f . This implies in particular that B h f = Bf + O ( h ) in L ph for any givensmooth function f with compact support. It is also classical that ( − ∆ h ) − is bounded in L ph uniformly w.r.t. h . This is the L p stability of the finitedifference scheme. The fundamental theorem of finite difference methodsimplies the L p convergence of the sequence of discrete second order Riesztransforms u h towards the continuous second order Riesz transform u . Discrete Riesz tranforms on Lie-Group.
Observe that the finite dif-ference Riesz transform u h = R i,h f h defined on the grid Ω h , also givesrise to a Riesz transform on the Lie group Ω = Z N . This is a conse-quence of the homogeneity of order zero of the Riesz transforms. Indeed,the equation ∆ h u h = ∂ i,h f h rewrites as ∆ u = ∂ i, f , where u ( y ) := u h ( y/h ), f ( y ) := f h ( y/h ) for all y ∈ Z N , and where ∆ and ∂ i, arethe discrete differential operators defined on Z N . We have also k u h k L ph = h N/p k u k L p and k f h k L ph = h N/p k f k L p . Notice that for all h , this ensuresthat k u h k L ph / k f h k L ph = k u k L p / k f k L p . Sharpness for Theorem 1.1 in the discrete setting.
In the continuoussetting, the sharpness was proved in [31] based on the combintation R α = R − R of second order Riesz tranforms. Let u ( k ) = R α f ( k ) a sequence ofsecond order Riesz transforms yielding the sharp constant C p in the estimate,that is k u ( k ) k p / k f ( k ) k p → C p as k goes to infinity. For each k ∈ N and h >
0, introduce the finite difference approximation f ( k ) h of f ( k ) and thecorresponding finite difference Riesz transform. Thanks to the convergenceof the finite difference scheme, we can extract a subsequence f ( k ) h k such that k u ( k )1 k p / k f ( k )1 k p = k u ( k ) h k k p / k f ( k ) h k k p → C p . Therefore C p is also the sharpconstant for the second order Riesz transforms in Z N . Sharpness for Theorem 1.3 in the discrete setting.
Recall that wehave a bound of the form k R α f k L p, ∞ ( G , C ) := sup E (cid:26) µ z ( E ) /p − Z E | R α f | d µ z (cid:27) C p k f k L p for a certain constant C p that is known to be sharp in the case of continuoussecond order Riesz transforms. In order to prove sharpness when the Liegroup G does not have enough continuous components, it suffices again toapproximate a sequence of continuous extremizers by a sequence of finitedifference approximations. Take G = R N . For any ε >
0, let f , u := R α f ,and E with finite measure chosen so that µ z ( E ) /p − k u k L ( E ) / k f k L p > C p − ε. We can assume without loss of generality that f is a smooth function withcompact support. Let f h a finite difference approximation of f defined as its L projection on the grid, and u h its discrete second order Riesz trans-form both defined on Ω h := h Z N . Since µ z ( E ) is the finite N –dimensionalLebesgue measure of E , we use outer measure approximations of E followedby approximations from below by a finite number of small enough cubes ofsize h centered around the grid points of Ω h , to define a “finite difference”approximation E h of E such that µ h ( E h ) := X x ∈ E h h N → µ z ( E )when h goes to zero. Since the discrete Riesz transforms are stable in L ,the Lax-Richtmyer theorem ensures that k u h k L h → k u k L which implies k u h k L h ( E ) → k u k L ( E ) and also k u h k L h ( E h ) → k u k L ( E ) . Therefore for h small enough, µ h ( E h ) /p − k u h k L h ( E h ) / k f h k L p > C p − ε. Let as before u ( y ) := u h ( y/h ), f ( y ) := f h ( y/h ) for all y ∈ Ω := Z N ,and E := E/h . We have sucessively µ h ( E ) = h N µ ( E ), k u h k L h ( E h ) = h N k u k L h ( E ) and k f h k L p = h N/p k f k L p . This yields immediately µ ( E ) /p − k u k L h ( E ) / k f k L p > C p − ε, allowing us to prove sharpness for the class of discrete groups we are inter-ested in. References [1] Nicola Arcozzi.
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